Resource allocation technique

ABSTRACT

An improved resource allocation system comprising a reliability decision engine ( 323 ), which allocates the portfolio&#39;s assets as required for the desired reliability portfolio. The reliability decision engine including two reliability decision engines, a basic reliability decision engine ( 325 ) and a robust reliability decision engine ( 327 ). The use of robust optimization makes it possible to determine the sensitivity of the optimized portfolio. Scenarios can be specified directly by the user or automatically generated by the system in response to a selection by the user. Inputs ( 329, 331 ) are applied to basic the basic reliability decision engine ( 325 ) and inputs ( 311 ) are applied to robust reliability decision engine ( 327 ).

THE CROSS REFERENCES TO RELATED APPLICATIONS

This application claims priority from U.S. provisional patentapplication 60/480,097. Hunter. et al., Reliability decision engine,filed 20 Jun. 2003, and discloses further developments of techniqueswhich are the subject matter of PCT/US01/00636, Hunter. et al., Resourceallocation techniques. filed 9 Jan. 2001 and claiming priority from U.S.provisional application 60/175,261. Hunter. et al., having the sametitle and filed 10 Jan. 2000. The U.S. National Phase of PCT/US01/00636is U.S. Ser. No. 10/018,696, filed 13 Dec. 2001, which is herebyincorporated by reference into the present patent application for allpurposes. The present patent application contains the entire Backgroundof the invention from U.S. Ser. No. 10/018,696 and the DetailedDescription through the section titled Computation of the real optionvalue of the portfolio.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention concerns techniques for allocating a resource among anumber of potential uses for the resource such that a satisfactorytradeoff between a risk and a return on the resource is obtained. Moreparticularly, the invention concerns improved techniques for determiningthe risk-return tradeoff for particular uses, techniques for determiningthe contribution of uncertainty to the value of the resource, techniquesfor specifying risks, and techniques for quantifying the effects andcontribution of diversification of risks on the risk-return tradeoff andvaluation for a given allocation of the resource among the uses.

2. Description of Related Art

People are constantly allocating resources among a number of potentialuses. At one end of the spectrum of resource allocation is the gardenerwho is figuring out how to spend his or her two hours of gardening timethis weekend; at the other end is the money manager who is figuring outhow to allocate the money that has been entrusted to him or her among anumber of classes of assets. An important element in resource allocationdecisions is the tradeoff between return and risk. Generally, the higherthe return the greater the risk, but the ratio between return and riskis different for each of the potential uses. Moreover, the form taken bythe risk may be different for each of the potential uses. When this isthe case, risk may be reduced by diversifying the resource allocationamong the uses.

Resource allocation thus typically involves three steps:

-   -   1. Selecting a set of uses with different kinds of risks;    -   2. determining for each of the uses the risk/return tradeoff;        and    -   3. allocating the resource among the uses so as to maximize the        return while minimizing the overall risk.

As is evident from proverbs like “Don't put all of your eggs in onebasket” and “Don't count your chickens before they're hatched”, peoplehave long been using the kind of analysis summarized in the above threesteps to decide how to allocate resources. What is relatively new is theuse of mathematical models in analyzing the risk/return tradeoff. One ofthe earliest models for analyzing risk/return is net present value; inthe last ten years, people have begun using the real option model; bothof these models are described in Timothy A. Luehrman, “InvestmentOpportunities as Real Options: Getting Started on the Numbers”, in:Harvard Business Review, July-August 1998, pp. 3-15. The seminal work onmodeling portfolio selection is that of Harry M. Markowitz, described inHarry M. Markowitz, Efficient Diversification of Investments, secondedition, Blackwell Pub, 1991.

The advantage of the real option model is that it takes better accountof uncertainty. Both the NPV model and Markowitz's portfolio modelingtechniques treat return volatility as a one-dimensional risk. However,because things are uncertain, the risk and return for an action to betaken at a future time is constantly changing. This fact in turn givesvalue to the right to take or refrain from taking the action at a futuretime. Such rights are termed options. Options have long been bought andsold in the financial markets. The reason options have value is thatthey reduce risk: the closer one comes to the future time, the more isknown about the action's potential risks and returns. Thus, in the realoption model, the potential value of a resource allocation is not simplywhat the allocation itself brings, but additionally, the value of beingable to undertake future courses of action based on the present resourceallocation. For example, when a company purchases a patent license inorder to enter a new line of business, the value of the license is notjust what the license could be sold to a third party for, but the valueto the company of the option of being able to enter the new line ofbusiness. Even if the company never enters the new line of business, theoption is valuable because the option gives the company choices itotherwise would not have had. For further discussions of real optionsand their uses, see Keith J. Leslie and Max P. Michaels, “The real powerof real options”, in: The McKinsey Quarterly, 1997, No. 3, pp. 4-22, andThomas E. Copland and Philip T. Keenan, “Making real options real”, TheMcKinsey Quarterly, 1998, No. 3, pp. 128-141.

In spite of the progress in applying mathematics to the problem ofallocating a resource among a number of different uses, difficultiesremain. First, the real option model has heretofore been used only toanalyze individual resource allocations, and has not been used inportfolio selection. Second, there has been no good way of quantifyingthe effects of diversification on the overall risk.

Experience with the resource allocation system of U.S. Ser. No.10/018,696 has demonstrated the usefulness of the system, but has alsoshown that it is unnecessarily limited. It is an object of the inventiondisclosed herein to overcome the limitations of U.S. Ser. No. 10/018,696and thereby to provide an improved resource allocation system.

SUMMARY OF THE INVENTION

The object of the invention is attained in one aspect by a technique fordetermining the reliability of the returns from a user-selected set ofassets. The new technique determines the mean time to failure (MTTF)reliability of the set of assets, that is, the probability that one ormore assets belonging to the set will fail to provide the desiredminimum return indicated for the assets.

The object of the invention is attained in another aspect by a robustoptimization technique in which the optimization is done over a set ofuser-selected or defined scenarios. In a scenario, values which are usedin the optimization are defined to vary stochastically across a rangeand a probability is associated with the scenario. The scenarios canthus be set up to represent extreme conditions such as those seen incrises and the optimization optimizes the worst-case value of the set ofassets over the set of scenarios. Examples of the kinds of scenariosinclude scenarios which correspond to the historical returns data forthe assets in the set, scenarios in which certain assets become highlycorrelated, and scenarios based on outliers in the historical returnsdata.

The object of the invention is attained in a third aspect by a method ofoptimization which first uses MTTF reliability to select the assets inthe portfolio and then optimizes the portfolio. Optimization may be doneusing the techniques of U.S. Ser. No. 10/018,696 or robust optimizationtechniques. In the optimization, the user may specify optimizationsubject to a plurality of constraints that specify a probability thatthe set of assets yield a desired minimum return. Further, optimizationsmay be done on portfolios in which assets have negative weights or inwhich the combined weights are more than 1, thereby permittingoptimization of portfolios that involve shorted assets or leveragedassets. Constraints are also possible which restrict the sum of theweights of a subset of the assets and which limit the portfolio'sdownside risks. The method of optimization further permits selectionamong a number of objective functions and adjusting the objectivefunction by assigning a premium or discount to the real value of one ormore of the assets. Also permitted in the method is selection among anumber of modes of quantifying risk.

Other objects and advantages will be apparent to those skilled in thearts to which the invention pertains upon perusal of the followingDetailed Description and drawing, wherein:

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a flowchart of resource allocation according to the resourceallocation system described in U.S. Ser. No. 10/018,696;

FIG. 2 is a flowchart of operation of the improved resource allocationsystem disclosed herein;

FIG. 3 is a data flow block diagram for the improved resource allocationsystem;

FIG. 4 shows the top-level graphical user interface for the improvedresource allocation system;

FIG. 5 shows the probability distribution for the probability that thereturn from a single asset will exceed a minimum;

FIG. 6 shows the graphical user interface for the input analysis tool;

FIG. 7 shows the graphical user interface for the visualization tool;

FIG. 8 shows the graphical user interface for defining a scenario;

FIG. 9 shows the window that appears when RDE 323 has completed anoptimization;

FIG. 10 shows the graphical user interface for selecting an objectivefunction;

FIG. 11 is a block diagram of an implementation of the improved resourceallocation system;

FIG. 12 is the schema of the database used in the implementation; and

FIG. 13 shows the contents of assets and parameters tab 421.

Reference numbers in the drawing have three or more digits: the tworight-hand digits are reference numbers in the drawing indicated by theremaining digits. Thus, an item with the reference number 203 firstappears as item 203 in FIG. 2.

DETAILED DESCRIPTION

The following Detailed Description will begin by describing howtechniques originally developed to quantify the reliability ofmechanical, electrical, or electronic systems can be used to quantifythe effects of diversification on risk and will then describe a resourceallocation system which uses real option analysis and reliabilityanalysis to find an allocation of the resource among a set of uses thatattains a given return with a given reliability. Thereupon will bedescribed improvements to the resource allocation system including thefollowing:

-   -   The use of MTTF reliability to select a portfolio of assets to        be optimized using real option analysis;    -   The use of robust optimization in the resource allocation        system;    -   The use of multiple constraints in optimization;    -   The use of various kinds of constraints in the optimization; and    -   Modifications of the objective function used in the        optimization.

The objective function is the function used to calculate the real optionvalues of the assets; in the original resource allocation system, theonly available objective function was the Black-Scholes formula usingthe standard deviation of the portfolio to express the portfolio'svolatility. The descriptions of the improvements will includedescriptions of the graphical user interfaces for the improvements. Alsoincluded will be a description of an implementation of a preferredembodiment of the improved system.

Applying Reliability Techniques to Resource Allocation

Reliability is an important concern for the designers of mechanical,electrical, and electronic systems. Informally, a system is reliable ifit is very likely that it will work correctly. Engineers have measuredreliability in terms of the probability of failure; the lower theprobability of failure, the more reliable the system. The probability offailure of a system is determined by analyzing the probability thatcomponents of the system will fail in such a way as to cause the systemto fail. A system's reliability can be increased by providing redundantcomponents. An example of the latter technique is the use of triplecomputers in the space shuttle. All of the computations are performed byeach of the computers, with the computers voting to decide which resultis correct. If one of the computers repeatedly provides incorrectresults, it is shut down by the other two. With such an arrangement, thefailure of a single computer does not disable the space shuttle, andeven the failure of two computers is not fatal. The simultaneous or nearsimultaneous failure of all three computers is of course highlyimprobable, and consequently, the space shuttle's computer system ishighly reliable. Part of providing redundant components is making surethat a single failure elsewhere will not cause all of the redundantcomponents to fail simultaneously; thus, each of the three computers hasan independent source of electrical power. In mathematical terms, if thepossible causes of failure of the three computers are independent ofeach other and each of the computers has a probability of failure of nduring a period of time T, the probability that all three will fail in Tis n³.

The aspect of resource allocation that performs the same function asredundancy in physical systems is diversification. Part of intelligentallocation of a resource among a number of uses is making sure that thereturns for the uses are subject to different risks. To give anagricultural example, if the resource is land, the desired return is aminimum amount of corn for livestock feed, some parts of the land arebottom land that is subject to flooding in wet years, and other parts ofthe land are upland that is subject to drought in dry years, the wisefarmer will allocate enough of both the bottom land and the upland tocorn so that either by itself will yield the minimum amount of corn. Ineither a wet or dry year, there will be the minimum amount of corn, andin a normal year there will be a surplus.

Reliability analysis can be applied to resource allocation in a mannerthat is analogous to its application to physical systems. The bottomland and the upland are redundant systems in the sense that either iscapable by itself of yielding the minimum amount in the wet and dryyears respectively, and consequently, the reliability of receiving theminimum amount is very high. In mathematical terms, a given year cannotbe both wet and dry, and consequently, there is a low correlationbetween the risk that the bottom land planting will fail and the riskthat the upland planting will fail. As can be seen from the example, theless correlation there is between the risks of the various uses for agiven return, the more reliable the return is.

A System that Uses Real Options and Reliability to Allocate InvestmentFunds: FIG. 1

In the resource allocation system of the preferred embodiment, theresource is investment funds, the uses for the funds are investments invarious classes of assets, potential valuations of the asset classesresulting from particular allocations of funds are calculated using realoptions, and the correlations between the risks of the classes of assetsare used to determine the reliability of the return for a particularallocation of funds to the asset classes. FIG. 1 is a flowchart 101 ofthe processing done by the system of the preferred embodiment.Processing begins at 103. Next, a set of asset classes is selected(105), Then an expected rate of return and a risk is specified for eachasset class (107). The source for the expected rate of return for aclass and the risk may be based on historical data. In the case of therisk, the historical data may be volatility data. In other embodiments,the expected rate of return may be based on other information and therisk may be any quantifiable uncertainty or combination thereof,including economic risks generally, business risks, political risks orcurrency exchange rate risks.

Next, for each asset class, correlations are determined between the riskfor the asset class and for every other one of the asset classes (108).These correlations form a correlation matrix. The purpose of this stepis to quantify the diversification of the portfolio. Thereupon, thepresent value of a real option for the asset class for a predeterminedtime is computed (109). Finally, an allocation of the funds is foundwhich maximizes the present values of the real options (111), subject toa reliability constraint which is based on the correlations determinedat 108.

Mathematical Details of the Reliability Computation

In a preferred embodiment, the reliability of a certain average returnon the portfolio is found from the average rate of return of theportfolio over a period of time T and the standard deviation a for theportfolio's return over the period of time T. The standard deviation forthe portfolio represents the volatility of the portfolio's assets overthe time T. The standard deviation for the portfolio can be found fromthe standard deviation of each asset over time T and the correlationcoefficient ρ for each pair of asset classes. For each pair A,B of assetclasses, the standard deviations for the members of the pair and thecorrelation coefficient are used to compute the covariance for the pairover the time T, with cov(A,B)_(T)=ρ_(A,B)σ_(A,T)σ_(B,T). Continuing inmore detail, for a general portfolio with a set S of at least two ormore classes of assets, the portfolio standard deviation and theportfolio's rate of return can be written as:

$\sigma_{P,T}^{2} = {{\sum\limits_{A \in S}{\sum\limits_{\underset{B \neq A}{B \in S}}{x_{a}x_{b}\rho_{AB}\sigma_{A,T}\sigma_{B,T}}}} + {\sum\limits_{A \in S}{x_{A}^{2}\sigma_{A,T}^{2}}}}$$r_{P,T} = {\sum\limits_{A \in S}{x_{A}r_{A,T}}}$

Where:

-   -   σ_(P,T) is the standard deviation (or volatility) of the        portfolio over T periods of time;    -   r_(p,t) is the average rate of return of the portfolio over T        periods of time;    -   x_(A) is the fraction of portfolio invested in asset class A;    -   ρ_(A,B) is the correlation of risk for the pair of asset classes        A and B;    -   σ_(A,T) is the standard deviation of asset class A over T        periods of time;    -   r_(A,T) is the average rate of return of asset class A over T        periods of time; and    -   S is the set of asset classes.

We assume in the following that the portfolio P follows a normaldistribution with mean of r_(P,T) and with standard deviation ofσ_(P,T): N(r_(P,T), σ_(P,T)).

The reliability constraint α will thus be:Pr(x≧r _(min))≧α

1−Φ((r _(min) −r _(P,T))/σ_(P,T)))≧αwhere r_(P,T) and σ_(P,T) are replaced by their respective values fromthe equation above. The constraint can be estimated using the expression

$\left( {r_{\min} - {\sum\limits_{A \in S}{x_{A}r_{A,T_{A}}}}} \right)^{2} \leq {\delta^{2}{\sum\limits_{A \in S}{\sum\limits_{\underset{\;}{B \in S}}{x_{A}x_{B}\sigma_{AB}}}}}$where δ² is obtained from a using Simpson's rule. Details of thecomputation of δ will be provided later.

Computation of the Real Option Value of the Portfolio

The above reliability constraint is applied to allocations of resourcesto the portfolio which maximize the real option value of the portfolioover the time period T. The real option value of portfolio is arrived atusing the Black-Scholes formula. In the formula, T_(A) is the time tomaturity for an asset class A and x_(Ai) is the fraction of theportfolio invested in asset class A during the period of time i, whereT_(A) is divided into equal periods 0 . . . T_(A)-1.

To price a real option for an asset class A over a time T according tothe Black-Scholes formula, one needs the following values:

-   -   A, the current value of asset class A;    -   T, time to maturity from time period 0 to maturity;    -   Ex, value of the next investment;    -   r_(f), risk-free rate of interest;    -   σ, volatility        A=x_(A0)P        Ex=x _(A0) P(1+r _(min,A))^(T) ^(A)

For a period i, the value V_(A,i) of the real option corresponding tothe choice of asset class A at time i using the Black-Scholes formulais:

$\begin{matrix}{V_{A,i} = {{{\Phi\left( \frac{{\log\;\left( \frac{1}{\left( {1 + r_{\min,A}} \right)^{T_{A^{- i}}}} \right)} + {\left( {r_{f} + {0.5\;\sigma^{2}}} \right)\left( {T_{A} - i} \right)}}{\sigma\sqrt{T_{A} - i}} \right)}x_{A,i}P} -}} \\{\Phi\left( {\frac{{\log\;\left( \frac{1}{\left( {1 + r_{\min,A}} \right)^{T_{A^{- i}}}} \right)} + {\left( {r_{f} + {0.5\;\sigma^{2}}} \right)\left( {T_{A} - i} \right)}}{\sigma\sqrt{T_{A} - i}} - {\sigma\sqrt{T_{A} - i}}} \right)} \\{x_{A,i}{P\left( {1 + r_{\min,T_{A}}} \right)}^{T_{\; A^{- i}}}\exp\;\left( {- {r_{f}\left( {T_{A} - i} \right)}} \right)}\end{matrix}$

The above formula is an adaptation of the standard Black-Scholesformula. It differs in two respects: first, it does not assumerisk-neutral valuation; second an exponential term has been added to thefirst term of V_(A,i) and corresponds to the discounted value for a rateof return r_(a). With these two changes, the real option value is bettersuited to the context of asset allocation.

The allocation of the available funds to the asset classes thatmaximizes the real option value of the portfolio can be found with theoptimization program

$\underset{\underset{A \in S}{x_{A,i}}}{Max}{\sum\limits_{A \in S}{\frac{1}{T_{A} - i}\left( {\frac{V_{A,i}}{x_{A,i}} - V_{\min,A}} \right)x_{A,i}}}$the program being subject to reliability constraints such as the one setforth above.

Overview of the Improved Resource Allocation System

The following overview of an improved version of the resource allocationsystem described above begins with an overview of its operation,continues with an overview of flows of information within the system,and concludes with an overview of the user interface for the system. Theimproved resource allocation system uses two measures for thereliability of a portfolio of assets. The first of these is a measure of“mean time to failure” (MTTF) reliability; the second is a measure oftotal return reliability. In the improved system, MTTF reliability isused to determine the reliability of sets of assets. A portfolioconsisting of a set of assets that has sufficient MTTF reliability isthen optimized using constraints that may include a constraint based onthe total return reliability measure.

In allocating assets, the user can take into account realisticreal-world constraints based on investor risk preferences, shorting,leverage, asset class constraints, minimum investment thresholds, anddownside constraints and devise optimal portfolios that maximize upsidepotential while accounting for liquidity, reliability of data, andpremiums or discounts associated with non-normal behavior of data.Instead of the single objective function and volatility measure used inthe original system, the improved system permits the user to chooseamong a number of objective functions and volatility measures.

The improved asset allocation system further incorporates robustoptimization, i.e., optimization which recognizes inherent uncertaintyin data and stochastic variations in parameter estimates to come up witha robust, reliable portfolio based on a set of comprehensive scenariosspanning the realm of possibilities for the assets in the portfolio andthe portfolio itself.

Overview of Operation: FIG. 2

Flowchart 201 in FIG. 2 presents an overview of how a user of theimproved resource allocation system uses the system. If the flowchart201 of FIG. 2 is compared with the flowchart 101 of FIG. 1, it willimmediately be seen that the improved system offers the user many moreoptions. In the system of FIG. 1, the user could only specify a set ofasset classes in step 105; everything else was determined by the systemfrom information in the system about the asset classes. In particular,the only objective function available was the Black-Scholes formula andthe only volatility measure that could be employed in the Black-Scholesformula was the standard deviation for the portfolio's assets over timeT; moreover, only a single constraint could be employed in theoptimization of the weights of the portfolio's assets, and thatconstraint was required to be a reliability constraint based on thetotal return reliability.

As shown in FIG. 2, by contrast, steps 203 through 211 involve settingoptions for the optimization step 213, which performs operations whichcorrespond functionally to those set forth in steps 107-111 of FIG. 1.In step 203, the user can select from a number of formulas for computingthe real option values of the portfolio's assets, can input parametersfor the effect of taxes on the portfolio, and can select how the risk isto be defined in the calculation. In step 205, the user can select theinvestment horizon for the optimization, the desired minimum return, theconfidence level desired for the portfolio, and the expected averagerisk free rate over the investment horizon.

In step 207, the user can specify a previously-defined portfolio foroptimization or can select assets to be included in the portfolio to beoptimized. In step 209, the user can employ the new capabilities of theimproved system to analyze various aspects of the selected portfolio,including analyzing the portfolio for clustering of returns from theportfolio's assets (which increases the risk of the portfolio as awhole), analyzing the correlation matrix for the portfolio's assets, andanalyzing the mean-time-to-fail (MTTF) reliability of the returns on theassets in the portfolio.

Step 211 permits the user to specify the initial, maximum, and minimumallocations of the assets selected for the portfolio in step 209 and tospecify one or more constraints that must be satisfied by the assets inthe portfolio. These constraints will be explained in detail later. Step213, finally, does the optimization selected in step 203 using theparameters selected in step 205 on the portfolio selected in steps 207and 209 using the allocations and constraints specified in step 211. Fora given optimization, the user may save the input configuration that wasset up in steps 203-211 and use it as the basis for a furtheroptimization. In general, what the user inputs in steps 203-211 willdepend on what has been previously configured and what is required forthe present circumstances.

Overview of Information Flows in the Improved Resource AllocationSystem: FIG. 3

FIG. 3 is a block diagram 301 that provides an overview of the flows ofinformation in the improved resource allocation system. The informationis received in reliability decision engine 323, which allocates theportfolio's assets as required for the desired reliability of theportfolio. In the improved resource allocation system, reliabilitydecision engine 323 includes two reliability decision engines: basicreliability decision engine 325, which optimizes in the general mannerdescribed in U.S. Ser. No. 10/018,696, and robust reliability decisionengine 327 which optimizes according to scenarios provided by the user.As will be explained later, the use of robust optimization makes itpossible to determine the sensitivity of the optimized portfolio tostochastic variations in the input parameters used to compute theoptimized portfolio. Portfolios optimized using basic RDE 325 can befurther fine tuned using robust optimization. Alternatively, robustoptimization can be used from the beginning. Scenarios can be specifieddirectly by the user or automatically generated by the system inresponse to a selection by the user.

Inputs provided by the user to the RDE are shown at 303, 311, 329, and331. Inputs 329 and 331 may be applied to both reliability engines;inputs 303 are applied to basic RDE 325 and inputs 311 are applied torobust RDE 327. The inputs fall generally into two classes: inputs whichdetermine how RDE 329 performs its computations and inputs whichdescribe the constraints that apply to the optimization. To the formerclass belong inputs 305 and 329; to the latter belong inputs 307,313,317, and 331. All of these inputs will be described in detail in thefollowing. Optional reliability MTTF constraint 321 permits the user toselect the assets in a portfolio according to whether the portfolio withthe selected assets has a desired MTTF reliability. If the MTTFreliability is not what is desired, no optimization of the portfolio isdone and the user selects different assets for the portfolio.

Overview of the User Interface for the Improved Resource AllocationSystem: FIGS. 4, 6-7, 13

The top-level user interface for the improved resource allocation systemis shown in FIG. 4. It is a typical windowing user interface. The toplevel window 401 of the user interface has four main parts: portfolioselection portion 402, which the user employs to select a portfolio ofassets or of benchmarks; optimization portion 404, which providesparameters for the optimization of the portfolio of assets selected bythe user in portion 402, and portfolio analysis tools at 406. Moduleselection portion 408, finally, permits selection of other modules ofthe asset management system of which the improved asset allocationsystem is a component. Of these modules, the ones which are important inthe present context are the asset module, which accesses assets andinformation about them, and the Profiler™ module, which permits detailedanalysis of the behavior of sets of assets. The Profiler is the subjectof the PCT patent application, PCT/US02/03472, Hunter, System forfacilitation of selection of investments, filed 5 Feb. 2002.

Beginning with portfolio selection portion 402, at 415, the user selectsa period of time from which the data about the assets in the set ofassets to be optimized will be taken At 416, the user can choose amongways of specifying portfolios: by selecting from a list of assets 417 orbenchmarks 419, by selecting from a list of portfolios that are orderedby the user's clients, or by selecting from a list of named portfolios.The names of the portfolios are generated automatically by the improvedresource allocation system. The naming convention is [ClientInitials]_[Date]_[Time Horizon]_[Target Return]_[Additional Constraintsin short]. At 419 is shown a list of benchmarks from which a portfoliomay be formed; a benchmark is added to a portfolio by checking the boxto the left of the benchmark.

Once a portfolio has been selected, it can be analyzed using the toolsat 406. Input analysis tool 403 permits the user to do detailed analysisof the set of assets being analyzed. In a preferred embodiment, thekinds of detailed analysis available include extreme values for thereturn and standard deviation of an asset in the set, extreme dates forthe return and standard deviation, extremes in the correlation matrixfor the set of assets, and extreme dates for the correlation matrix.Visualization tool 405 permits the user to visualize clustering in themultivariate normal distribution for the portfolio. Correlation matrixtool 409 permits the user to see the correlation matrix for theportfolio. Reliability tool 411 permits the user to compute the MTTFreliability for the portfolio. Objective function selection tool 413permits the user to select one of a number of objective functions. Theselected function is then used in the optimization. Where further userinput is required after selection of one of these functions, selectionof the function results in the appearance of a window for the furtheruser input. This is illustrated in FIG. 6, which shows display 601 thatresults when input analysis tool 403 is selected. Window 603 appears andthe user selects the kind of analysis desired at 605. The result of theselected function appears in another window. Display 701 in FIG. 7 showswindow 703 which contains a graph 705 that shows clustering of returnsin the multivariate normal distribution for the portfolio. The windowappears when the user clicks on visualization tool 405.

The user provides additional information needed to do an optimization inoptimization portion 404. Optimization portion 404 has two main parts:Assets and parameters 421 permit the user to specify the investmenthorizon, the risk free rate, downside risk options, whether returns aretaxable or not, tax rates if applicable, and automatic extraction of taxrates for the account information for the account for which theoptimization is being performed. The interface 1301 that appears whenthe user clicks on assets and parameters tab 421 is shown in FIG. 13. At1303, the user specifies the risk-free rate of return that is expectedduring the investment horizon for which the optimization is beingperformed. At 1305, the user specifies the investment horizon, i.e., theperiod of time for which the optimization is being performed. At 1307,the user inputs tax information for the account for which theoptimization is being done. Included are whether the returns are taxableand the account's tax rates for long term gains, short term gains, anddividends. At 1309, the user selects one of three modes of quantifyingdownside risk: whether it is uniform at −10% for all assets, whether itis based on the standard deviation, or whether it is based on the worstannual rolling returns for the assets. At 1311 are listed the assetsthat make up the portfolio together with statistics concerning theasset's return. Checkboxes in the rightmost column permit the user toindicate whether the asset's returns are taxable.

Optimization part 423 permits the user to input constraints on theoptimization such as the targeted return on the portfolio at 425, thelevel of confidence that the portfolio will provide the targeted returnat 426, and additional constraints at 427. At 429, the user may inputrobust optimization scenarios for use when the user has selected anobjective function that does robust optimization. At 431 is a list ofthe assets in the portfolio; using the list, the user can specifyallocation constraints including a maximum, minimum, and initialallocation for each asset in the portfolio; the user can also indicatewhether an asset may be “shorted”, i.e. borrowed from a willing lender,sold for a price A, and then purchased for a price B which is hopefullylower than A, and returned to the lender. Since a shorted asset is“owed” to the lender, the shorted asset's minimum allocation for theportfolio may be negative.

Once all of the information needed for the optimization has beenentered, the user clicks on run optimization button 433 to begin theoptimization. The asset allocation system then runs until it hasproduced an optimized portfolio which to the extent possible conforms tothe constraints specified by the user. FIG. 9 shows graphical userinterface 901 with the results of an optimization. Optimization resultwindow 903 has three main parts: list 909 of the assets in theportfolio, with the optimal weight of each asset. Note that the optimalweight for some of the assets is 0. At 905 are listed parameters used inthe optimization and at 907 are shown the results of the optimizationfor the portfolio as a whole. Of particular interest in the results arethe uncertainty cushion and catastrophic meltdown scenario, both ofwhich will be described later, and the list of confidence levels for arange of different rates of return.

If the user believes the optimized portfolio is worth saving, the userpushes save run button 435 which saves the optimized portfolio resultingfrom the run and the information used to make it. The optimizedportfolio can then be further analyzed using the improved resourceallocation system. For example, once a satisfactory optimized portfoliohas been obtained using basic RDE 325, scenarios of interest and theirprobabilities can be specified and the optimized portfolio can be usedas a scenario in robust optimization. A saved portfolio can also beperiodically subjected to MTTF analysis or reoptimization using currentdata about the returns and/or risks for the asset to determine whetherthe portfolio's assets or the assets'weight in the portfolio should bechanged.

Selecting a Set of MTTF-reliable Assets

Definitions and Assumptions

The following discussion uses the following definitions and assumptions:

Definition of an Asset

Initially, an asset A is simply defined as an entity whose returnsfollow a normal distribution. Thus each asset is represented by its meanand the variance. This is a fundamental assumption of several techniquesin finance theory, and is necessary for and consistent with theassumptions used in the Black-Scholes option valuation technique. In thefollowing theoretical discussion, this is the only assumption that wewill make about the nature of the asset.

Assumption Concerning the Return on an Asset

We initially assume that the return on an asset {tilde over (r)}_(A) isa normally distributed random variable.{tilde over (r)} _(A) ≈N(r _(A),σ_(A) ²)

While this assumption may not be valid for all assets, we see that forassets with a history more than 3-4 years, the asset returnsdistribution is pseudo-normal.

The normal distribution has a property that it can be completelydescribed by two parameters: its mean and variance, which arerespectively, the first and second moments of the asset returnsdistribution. When a random variable is subject to numerous influences,all of them independent of each other, the random variable isdistributed according to the normal distribution. The randomdistribution is perfectly symmetric—50% of the probability lies abovethe mean. For the normal distribution, the probability of the randomvariable lying within the limits of (m−s) and (m+s) is 68.27 % andwithin (m−2s) and (m+2s) is 95.45 %.

Measuring the Reliability of a Portfolio

In U.S. Ser. No. 10/018,696, the reliability of a portfolio of weightedassets was measured in terms of the probability that the portfolio willyield a desired minimum return r_(MIN). When the portfolio wasoptimized, the constraint under which the portfolio was optimized wasthat the probability that r_(MIN) would yield a given minimum return begreater than α. In the following, this measure of reliability is termedtotal return reliability. In the improved asset allocation system, anadditional measure of reliability is employed: mean time to fail (MTTF)reliability. The MTTF reliability of a set of assets is the probabilitythat during a given period of time one or more of the assets in the setwill not provide the minimum return desired for the asset.

It should be noted here that the MTTF reliability of a set of assets isindependent of the weight of the assets in the set and can thus be usedas shown at 321 in FIG. 3 to validate the selection of the set of assetsmaking up a portfolio prior to optimizing the portfolio. An importantfeature of the improved asset allocation system is that it includes sucha selection validator 321 in addition to RDE optimizer 323. Thefollowing discussion will show how the MTTF reliability for a set ofassets is computed and how the computation is used in the improved assetallocation system. The total return reliability will be discussed indetail along with the other constraints used in optimization.

We will begin the discussion of MTTF reliability by showing how themultivariate normal distribution for a portfolio can be used todetermine the probability that each asset in a portfolio will perform,i.e., meets a desired minimum return on the asset.

Using the Multivariate Normal Distribution to Determine the Probabilitythat an Asset will Perform: FIG. 5

Let U be the universe of such assets A, B, C . . . N.

We know that ∀Asset A ∈ Universe U

{tilde over (r)}_(A)≈N(μ_(A),σ_(A) ²)

${{{Let}\mspace{14mu}\overset{\sim}{R}} \equiv \begin{bmatrix}{\overset{\sim}{r}}_{A} \\{\overset{\sim}{r}}_{B} \\{\overset{\sim}{r}}_{C} \\\vdots \\{\overset{\sim}{r}}_{N}\end{bmatrix}},$be the random variable associated with the portfolio returns μ≡E[{tildeover (R)}], the mean of the portfolio returns and V≡Var({tilde over(R)}), the variance of the portfolio returns

Therefore the multivariate normal distribution is given by:

${\overset{\sim}{R} \approx {N_{{{Universe}\mspace{14mu} U}}\left( {\mu,V} \right)}},{where}$$\mu = {\begin{bmatrix}{E\left\lbrack {\overset{\sim}{r}}_{A} \right\rbrack} \\{E\left\lbrack {\overset{\sim}{r}}_{B} \right\rbrack} \\{E\left\lbrack {\overset{\sim}{r}}_{C} \right\rbrack} \\\vdots \\{E\left\lbrack {\overset{\sim}{r}}_{N} \right\rbrack}\end{bmatrix} = {\begin{bmatrix}\mu_{A} \\\mu_{B} \\\mu_{C} \\\vdots \\{\mu_{N}}\end{bmatrix}\mspace{14mu}{and}}}$ $V = \begin{bmatrix}\sigma_{A}^{2} & {\rho_{A,B}\sigma_{A}\sigma_{B}} & \cdots & {\rho_{A,N}\sigma_{A}\sigma_{N}} \\{\rho_{A,B}\sigma_{A}\sigma_{B}} & \sigma_{B}^{2} & \cdots & {\rho_{B,N}\sigma_{A}\sigma_{N}} \\\vdots & \vdots & ⋰ & \vdots \\{\rho_{A,N}\sigma_{A}\sigma_{N}} & {\rho_{B,N}\sigma_{B}\sigma_{N}} & \cdots & \sigma_{N}^{2}\end{bmatrix}$

{tilde over (R)} is a random vector of portfolio returns. Since {tildeover (R)} is a function of N random variables, each following a normaldistribution, {tilde over (R)} follows a multivariate normaldistribution.

The justification for construction of the multivariate normaldistribution is as follows. From the universe of possible assets U, letus identify a subset Q (Q⊂U) of assets upon which we wish to place anadditional constraint. Consider an investor who, for each asset Abelonging to Q, requires that the return on that asset be above athreshold minimum return r_(min,A). Since the asset returns in Q arejointly normally distributed, it is possible to ex ante calculate theprobability of this event occurring.

Illustrating this constraint when Q contains a single asset X is easy.As just shown, our chosen asset X has returns {tilde over (r)}_(x) thatare normally distributed with mean μ_(x) and variance σ_(x) ². There areno constraints on any other asset in U. Therefore, the only relevantasset return distribution to consider is the distribution of assetreturn {tilde over (r)}_(x), which is depicted in FIG. 5. Because thereturns are normally distributed, they form a bell curve 503. Line 505shows the minimum desired return. The probability that {tilde over(r)}_(x) exceeds r_(min,x), Pr({tilde over (r)}_(x)>r_(min,x)), isrepresented by the area of shaded portion 507. Let us call theprobability represented by shaded portion 507 probability p. Elementaryprobability gives us the value of p; it is simply

${\Phi\left( \frac{\mu_{X} - r_{\min,X}}{\sigma_{X}} \right)},$the value associated with the cumulative distribution of asset X atr_(min,X).

Let us now return to our investor in order to understand thesignificance of this calculation for asset allocation systems like theone disclosed here and the one disclosed in U.S. Ser. No. 10/018,696,which will be termed in the following real option value asset allocationsystems. At the simplest level, p is exactly what we defined it tobe—the probability of the return on asset X exceeding the minimum returnon that asset. But this same number has other meanings. In real optionvalue asset allocation systems, p also gives us the probability that areal option drawn on asset X is “in-the-money” at the end of the optionperiod. This probability is important because real option value assetallocation systems only value future states of the world where thereturn on an asset is equal to or exceeds the minimum return on thatasset. Put another way, real option value asset allocations systemsfavor options that will be “in-the-money” and thereby maximize upsidepotential. Future states of the world in which assets perform belowminimum are not valued, and do not contribute to the asset weights usedduring optimization.

Thus, the probability that an investment in asset X “performs”, or is“in-the-money” gives the user of a real option value asset allocationsystem a value which can be used to validate the asset weights used inthe optimization. As will be seen later, it can also be used toconstruct a measure of reliability for a set of assets.

In order to build intuition, let us extend this example to case whenQ={X, Y}, but restrict ourselves to the improbable scenario where {tildeover (r)}_(X) and {tilde over (r)}_(Y) are uncorrelated and henceindependent. The probability that the minimum return criterion is metfor both asset returns is given by the expression Pr({tilde over(r)}_(X)>r_(min,X))·Pr({tilde over (r)}_(Y)>r_(min,Y)|{tilde over(r)}_(X)>r_(min,X)). Since {tilde over (r)}_(X) and {tilde over (r)}_(Y)are independent, the conditional probability expression Pr({tilde over(r)}_(Y)>r_(min,Y)|{tilde over (r)}{tilde over (r_(X))}>r_(min,X))collapses to the simpler expression PR({tilde over (r)}_(Y)>r_(min,Y)).Hence the probability that the minimum return criterion is met for bothasset returns is given by the expression

${\Phi\left( \frac{\mu_{X} - r_{\min,X}}{\sigma_{X}} \right)} \cdot {{\Phi\left( \frac{\mu_{Y} - r_{\min,Y}}{\sigma_{Y}} \right)}.}$This is similar to the expression derived in the first example.

Unfortunately, the elegance of this solution is based upon theunrealistic assumption of independence amongst asset returns. In thegeneral case, correlations amongst asset returns are significant and maynot be ignored in this fashion.Let U={A, B, C . . . M}, with correlated asset returnsLet p=Pr({tilde over (r)}_(A) >r _(min,A)AND {tilde over (r)} _(B) >r_(B)AND, . . . {tilde over (r)}_(M) >r _(min,M))

In the general case,

p = ∫_(r_(min , M))^(∞)⋯∫_(r_(min , C))^(∞)∫_(r_(min , B))^(∞)∫_(r_(min , A))^(∞)f_(U)(a, b, c  …  m) ∂a∂b∂c  …∂m

In the above equation, ƒ_(|Q|)(•) is the probability density functionfor a multivariate normal distribution. Thus p is the probability thateach of the selected assets meet its desired minimum return in theinvestment period. Since each of these normally distributed assets iscorrelated, the returns on the portfolio as a whole obey themultivariate normal distribution. Therefore the probability that eachasset in the selected set ‘performs’ i.e. meets the desired minimumreturn on that asset is the value associated with the multivariatecumulative distribution of portfolio returns evaluated at the desiredminimum returns, given by p in the above equation.

Using p to Compute the MTTF Reliability of a Portfolio

p can be used to compute the MTTF reliability of a portfolio of assets.Under the normality assumption, the ex ante probability distribution of{tilde over (r)}_(X) is a normal distribution as shown in FIG. 5. Shadedarea 507 gives us the region where {tilde over (r)}_(X) exceeds theminimum return. Area 507 may also be interpreted as the number of allpossible future outcomes in which the minimum return constraint is met.Since the objective function assigns weights to the portfolio's assetsunder the assumption that the strike price of the asset option is theminimum return, area 507 is proportionate to the total number of futureoutcomes in which the construction of the objective function isaccurate. Let this number be n(T) Now, let n₀(T) denote the total numberof possible future outcomes. In this case, the reliability of theobjective function reduces to n(T)/n₀(T)=p.

Because this is so, p is also a reliability measure for the objectivefunction. Validator 321 determines p for a given set of assets and agiven period of time. Since p is the probability that each of the assetswill perform in the given period and the mean-time-to-failurereliability (MTTF) for a given period of time for the portfolio is theprobability that one or more of the assets will not perform during thegiven period of time,MTTF=1−p

Using Validator 321 to Select Assets for a Portfolio

Validator 321 works as follows: the user selects a set of assets usingselection part 402 of the graphical user interface and then clicks onMTTF tool button 411. The asset allocation system responds to thoseinputs by computing the MTTF reliability of the set of assets. Thereliability of the set is 1−p, and the value of that expression appearsas a percentage on button 411 in the place of the question marks thatare there in FIG. 4. For example, if p has the value 0, 100% appears onbutton 411.

Efforts were made to optimize the selection of the assets themselves.The idea was to come up with a set of assets with an optimal MTTFreliability and to then optimize the weights of the assets in aportfolio made up of the set or assets. However, the optimization forMTTF reliability has an exponential running time. Say we have n assetsto choose from. The number of possible sets with these n assets would be2^(n). Moreover, since these are discrete states, we cannot devise anintelligent way to traverse these sets to get the optimal set. Giventhat the running time for optimizing MTTF reliability is exponential, itis much more efficient to allow the user to select the assets in theallocation and have the system determine the MTTF reliability of theselected set. Once the user is satisfied with the MTTF reliability of aset of assets, he then uses optimization part 404 of the user interfaceto optimize the weights of the assets in the portfolio made up of theset with the satisfactory MTTF reliability.

Robust Optimization

Introduction

In optimization as performed by basic reliability decision engine 325,the optimization has the following characteristics:

-   -   The real option value of a portfolio of assets is maximized        subject to constraints of non-linear reliability, upper and        lower bounds on each asset and upper and lower bounds on linear        combinations of assets, with or without shorting and with or        without leverage.    -   The objective function and the constraints are computed using        the means and covariances provided by historical asset returns

A necessary limitation of this kind of optimization is that these meansand covariances are historical. They describe past behavior of theassets over relatively long periods and by their very nature cannotdescribe the behavior of the assets in times of crisis. For example, intimes of crisis, assets that bear a low correlation with the broadindices and with each other in normal times, have been known to gethighly correlated. Further, times of crisis are normally associated witha serious liquidity crunch and the crunch occurs just at the time whenall asset correlations rapidly grow towards 1.

Robust optimization deals with the fact that it is uncertain whether thehistorical trends for an asset or a set of assets would continue intothe future. Robust optimization has its origins in control systemsengineering. The aim of robust optimization is to take into accountinherent uncertainties in estimating the average values of the inputparameters when arriving at an optimal solution in a system which in ourcase is defined by a set of non-linear equations. Where the standardoptimization program takes an individual parameter as input, the robustoptimization program expects some measure of central tendency for theinput parameter and a description of stochastic variation of the actualinput parameter from that measure. In the context of the optimizationdone by RDE 323, this approach is applied to the mean, standarddeviation and correlations which serve as parameters for theoptimization. Thus, in the optimization performed by robust RDE 327, anadditional input is added, namely, a measure of the stochastic variationassociated with the mean, standard deviation, and correlation parametersdescribing the returns distribution. Of course, the same constraints canbe used with the robust optimization performed by RDE 327 as with thebasic optimization performed by RDE 325.

It is important to note that the notions of reliability and robustnessare orthogonal to each other. In the context of RDE 323, reliability isa check on the validity of the constructed objective function whereasrobustness is a measure of the sensitivity of the optimization output tostochastic variations in the input parameters.

Details of Robust Optimization in the Improved Resource AllocationSystem

Scenarios for Robust Optimization

Robust RDE 327 performs robust optimization of a set of assets on thebasis of a set of possible extreme scenarios. Each scenario is describedusing the mean return, μ, and the covariance matrix Σ for the set ofassets. Each of the extreme scenarios also includes a probability of thescenario's occurrence. Robust RDE 327 maximizes the worst-case realoption value of a portfolio of assets over the set of scenarios, eachwith a given probability of occurrence. The objective function for therobust optimization performed by RDE 327 is:

${\underset{\underset{W}{︸}}{Maximize}\mspace{11mu}\underset{\underset{\mu,{\sum{\in {{S\ldots}\; 1\text{:}k}}}}{︸}}{Min}{\sum\limits_{i}\left( {v_{i}^{T} \cdot x_{i}} \right)}},$where v_(i) and x_(i) are the adjusted real option value and theallocation to asset i respectively and set

$S = \left\{ {{\sum{\in {{R^{n \times n}\text{}\sum} \succcurlyeq 0}}},{\sum\limits_{\underset{i,j}{\_}}{\leq {\sum\limits_{i,j}\overset{\_}{\sum\limits_{i,j}}}}}} \right\}$is comprised of scenarios 1 through k, the total number of independentscenarios and covariance matrix Σ is positive semi-definite and boundedsubject to the two stochastic variation constraints:

${\underset{\_}{\mu_{i}} \leq \mu_{i} \leq \overset{\_}{\mu_{i}}}\mspace{14mu}$i = 1, …  n  and${\sum\limits_{\underset{i,j}{\_}}{\leq {\sum\limits_{i,j}{\leq {\overset{\_}{\sum\limits_{i,j}}i}}}}},{j = 1},\ldots\mspace{11mu},n,$where the estimate of the mean return for an asset and elements of thecovariance matrix lie between two extremities given by the stochasticvariation of the mean and covariance respectively.

The above optimization problem is convex overall and RDE 327 solves itusing the techniques and algorithms of conic convex programmingdescribed by L. Vandenberghe and S. Boyd in SIAM Review ( 38(1):49-95,March 1996) and software for convex SCONE programming available as ofJune, 2004 through S. Boyd at www.stanford.edu/˜boyd/SOCP.html

The Interface for Defining Scenarios: FIG. 8

In a preferred embodiment, the user defines scenarios for a particularset of assets. The user can specify properties for a scenario asfollows:

-   -   the desired performance for the scenario;    -   the probability of the scenario's occurrence;    -   the downside risk for the scenario; and    -   how the correlation between the assets is to be computed.

FIG. 8 shows the user interface 801 for doing this. The set of windowsshown at 803 appear when the user clicks on “Input robust optimizationscenarios” button 429. At 805 are seen a drop-down list of scenarios,with the name of the scenario presently being defined in field 806 and aset of scenario editing buttons which permit the user to add a scenario,update the assets to which the scenario in field 806 applies, and deletethat scenario. The assets for the scenario specified in box 806 areshown in list 815.

Windows 807, 815, and 817 contain current information for the scenariowhose name is in field 806. The fields at 809 permit the user to specifyassumptions for the scenario including the risk-free interest rate, theinvestment horizon, the desired portfolio return, correlations betweenthe assets, and the desired confidence level for the portfolio. At 810,the user inputs the probability of the scenario. The user employs thebuttons at 811 to select the downside risk the optimizer is to use inits calculation and the buttons at 811 to select the source of thevalues for the correlation matrix to be used in its calculation.

The buttons in correlation computation 813 permit definition of thefollowing types of scenarios in a preferred embodiment:

-   1) A scenario where means and covariance between assets are equal to    parameters calculated from historical data. This scenario is the one    corresponding to the optimization done by basic RDE engine 325.-   2) A scenario in which the covariance matrix is estimated from    outliers in the asset returns. This may better characterize the    “true” portfolio risk during market turbulence than a covariance    matrix estimated from the full sample.

The user may set up his own scenario in which correlations between allor some assets become 1, i.e. assets get highly correlated by inputtingsuch correlations to the correlation matrix for the set of assets (meanreturns may be assumed to be equal to historical mean returns). Theability to handle means and covariances for other types of scenarios maybe incorporated into robust RDE 327.

One example of another type of scenario is the following: If we are ableto forecast the mean/covariance matrix for some assets, each set of suchforecasts would potentially constitute a scenario. Forecasts of returnsbased on momentum, market cycle, market growth rates, fiscal indicators,typical credit spreads etc. could be used for scenarios, as couldforecasts of the risk free rate, drawdown etc. of specific assets. Theforecasts can be obtained from external forecasting reports.

In addition to using different sources for the means and covariances inthe scenarios that the robust optimizer is optimizing over, it is alsopossible to use different objective functions in different ones of thescenarios, with the objective function employed with a particularscenario being the one best suited to the peculiarities of the scenario.

Maximizing the worst-case real option value of the portfolio of assetsfor all scenarios defined for a portfolio may not be suited for allapplications. One situation where this may be the case is if one or moreof the scenarios has a very small probability of occurrence. Anothersuch situation is when the scenarios defined for the portfolio includemutually exclusive scenarios or nearly mutually exclusive scenarios. Todeal with this, the defined scenarios can be divided into sets ofmutually-exclusive or nearly mutually-exclusive scenarios and theprobability of occurrence specified for each of the scenarios in a set.The robust objective function could then maximize on the basis of theprobabilities of occurrence of the scenarios of a selected set.

Scenario Generation Using Outliers

A button in correlation computation area 813 permits the user to specifyoutliers in the historical returns data as the source of the correlationmatrix for the portfolio. Robust RDE 327 then correlates an outliercorrelation matrix as follows:

In a preferred embodiment of RDE 323, the correlation matrix isordinarily computed using a “cut-off” of 75% meaning that if the set ofreturns falls beyond the cut-off point in the n-dimensional ellipsoid,it is treated as an outlier. The set of returns used to compute thecorrelation matrix is defined as the n-dimensional ellipsoidal set

$\underset{\underset{k}{︸}}{R} = {r^{k}\left\{ {r_{1},r_{2},\ldots\mspace{11mu},r_{n}} \right\}}$where n denotes the number of assets in the portfolio and k denotes thenumber of common data points available for the n assets.

When the outlier correlation matrix is being computed, the “cut-off” isused to calculate a composite measure ζ, inverse chi-square valueassociated with a chi-square distribution characterized by the cut-offvalue and n degrees of freedom, where n is the number of assets. Now,the outlier-correlation matrix is constructed based on a subset S of thek data points

$\begin{matrix}{{S = {{r^{s}\left\{ {r_{1},r_{2},\ldots\mspace{14mu},r_{n}} \right\}{s.t.\mspace{14mu}{\mathbb{d}{t\left( r^{s} \right)}}}} \geq \zeta}},\mspace{14mu}{{where}\mspace{14mu}{\mathbb{d}t}\mspace{14mu}{is}\mspace{14mu}{given}\mspace{14mu}{by}}} \\{{\mathbb{d}t} = {{\left( {r^{k} - \mu} \right)^{T} \cdot {\sum\limits^{- 1}{{\cdot \left( {r^{k} - \mu} \right)}r^{k}\left\{ {r_{1},r_{2},\ldots\mspace{14mu},r_{n}} \right\}}}} \in R}}\end{matrix},$Σ is the covariance matrix for the given scenario and μ is the vector ofestimates for mean returns on the assets. As can be seen, S⊂R, i.e. Swould be a subset of R.

Doing Robust Optimization

In a preferred embodiment, the user selects robust optimization or basicoptimization when the user selects the objective function for theoptimization. The user interface for doing this is shown in FIG. 10,described below.

Constraints Employed in the Improved Resource Allocation System

The Total Return Reliability Constraint

This constraint is employed in the improved resource allocation systemin the same fashion as in the system of U.S. Ser. No. 10/018,696. It isused in all optimizations done by basic RDE 325 and is one of thecorrelation computations that may be used to define a scenario in robustoptimization.

The formula for this constraint is derived as follows: Consider anallocation vector

${\overset{\rightarrow}{x} \equiv \begin{bmatrix}x_{A} \\x_{B} \\x_{C} \\\vdots \\x_{N}\end{bmatrix}},$where x_(A) is the proportion of the portfolio invested in asset A.

If {tilde over (P)} is the return on a portfolio allocation with weights{right arrow over (x)}, then

$\overset{\sim}{P} \equiv {{\overset{\rightarrow}{x}}^{T}\overset{\sim}{R}} \approx {N\left( {r_{P},\sigma_{P}^{2}} \right)}$$r_{P} = {\sum\limits_{A \in U}{x_{A}r_{A}}}$$\sigma_{P}^{2} = {\sum\limits_{A \in U}{\sum\limits_{B \in U}{\rho_{A,B}\sigma_{A}\sigma_{B}}}}$

If we place the constraint that the probability that the portfolioyields a desired minimum return r_(MIN) is greater than a desiredconfidence level α,Pr({tilde over (P)}>r _(MIN))>α, Then:

$\left. {{\Pr\left( {\overset{\sim}{P} > r_{MIN}} \right)} > \alpha}\Rightarrow{r_{MIN} < {\left( {1 - \alpha} \right){quantile}\mspace{14mu}{of}\mspace{14mu}\overset{\sim}{P}\mspace{14mu}{distribution}}}\Rightarrow{{\Phi\left( \frac{r_{MIN} - r_{P}}{\sigma_{P}} \right)} < \left( {1 - \alpha} \right)}\Rightarrow{\frac{r_{P} - r_{MIN}}{\sigma_{P}} > {\Phi^{- 1}(\alpha)}} \right.$

The total return reliability constraint ensures that the probabilitythat the ‘returns on the portfolio’ exceed the ‘minimum desired returnon the portfolio’ is greater than a confidence level α. If thatconfidence level is not achievable by the selected set of assets for thedesired minimum return on the portfolio, then RDE 323 optimizes around a5% interval around the peak confidence achievable by the selected setfor the given desired minimum portfolio return.

User Interface for Defining Constraints: FIG. 4

FIG. 4 shows the user interface used in a preferred embodiment fordefining constraints other than the total return reliability constraintat 431. Each asset has a row in the table shown there, and columns inthe rows permit definition of the constraints that are explained indetail in the following.

Details of the User-Defined Constraints

Constraints Permitting Shorting and Leverage of Assets

The RDE, in its most basic optimization version, assumes no leverage orshorting, which means that the weights of all the assets in theportfolio are all non-negative and sum up to 1.No Shorting 0≦x_(i)≦1No Leverage Σ(x _(i))=1

However, the advanced version of the RDE allows both shorting andleverage.

Shorting

When shorting is allowed, the minimum allocation for an asset may benegative. The previous non-negativity constraint in the optimizationalgorithm is relaxed for any asset in which it is possible or desirableto take a short position. Thus, the weight of an asset in a portfoliomay range betweens≦x_(i)≦l,where s and l can be negative, positive or zero. Typically, s would notbe less than −1 and l not grater than +1, but theoretically, they cantake values beyond −1 and 1.

Also, for the short asset the real-option value may be computed usingthe negative of the mean return for the asset, with the same standarddeviation as the long asset. However, while assessing the downside riskof the short asset, the best performing 1-year rolling period of thelong asset must be considered as a gauge of the worst-possible downsidefor the short asset. Alternatively, a maximum annualized trough to peakapproach can be used as a downside measure.

Leverage

When leverage is allowed, the sum of the asset allocation can exceed 1i.e. 100%. The Σ(x_(i))=1 constraint for the weights of the assets inthe portfolio would no longer be valid. Instead, the maximum on the sumof allocations would be governed by the leverage allowed.S≦Σ(x _(i))≦L,where S and L are determined by the maximum leverage allowed on theshort side and long side.

For example, if maximum allowable leverage is 2× or 200%, then the Lwould take a value of 2. In case we do not want the portfolio to be netshort, S would take a value of zero. Additionally, if we have to be atleast 30% net long with a maximum allowable 1.5× leverage, then S=0.3and L=1.5.

Multiple Asset Constraints

Constraints that specify restrictions on groups of assets may also beemployed in RDE 323. For example, the user is able to specify aconstraint that the sum of specific assets in the portfolio should havea necessary minimum or an allowable maximum. Any number of suchconstraints may be added to the optimization, allowing us to arrive atpractical portfolios that can be implemented for a particularapplication.

Also, if we allow selling securities/assets short, resources accumulatedby selling-short one asset can be used to buy another asset. Thereby theweight of the asset/s that has been short-sold will be negative and theweights of some of the other assets may even be greater than one. Asimilar situation might occur when allowing leverage as described in theprevious section.

Minimum Allocation Thresholds Constraint

Some assets have a minimum investment threshold which makes anyallocation below a specified dollar amount unacceptable. This can bemodeled as a binary variable that takes a value zero when the optimalallocation (from the non-linear optimization) is less than the minimumthreshold equivalent to the minimum allowable dollar investment in theasset. Such an approach pushes the optimization into the realm of mixedinteger non-linear programming wherein we use a branch-and-boundapproach that solves a number of relaxed MINLP problems with tighter andtighter bounds on the integer variables. Since the underlying relaxedMINLP model is convex, the relaxed sub-models would provide valid boundson the objective function converging to a global optimum, giving anallocation that accounts for minimum allocation thresholds for the givenset of assets.

Modeling Portfolio Return Reliability with Multiple a Constraints

The total return reliability constraint ensures that the probability ofportfolio returns exceeding a minimum desired return is greater than aspecified confidence level α. However, it is also possible to model thecomplete risk preference profile of the investor using multipleportfolio confidence constraints. For example, if an investor cannottolerate a return below 8% but is satisfied with a portfolio with a 60%probability of yielding a return over 12%, then we can model this riskaversion using two return reliability constraints:

-   -   Probability of minimum 8% return should be very high, say 99%    -   Probability of minimum 12% return should be 60%

In the optimization, while inching towards the optimal solution, we makesure that the most limiting return reliability constraint is consideredat every iteration. The most limiting constraint is calculated bycomparing the values of the specified return reliability constraints ateach iteration. Thus the most limiting constraint might change from oneiteration to another. Once the most limiting constraint is satisfied,all the other confidence constraints are recomputed to check if theyhave been satisfied. This is coded in Matlab as a separate constraintfunction. The optimization moves back and forth between the constraintsat each iteration, changing the most limiting constraint but slowlyinching towards the optimal solution satisfying all these confidenceconstraints.

Catastrophic Meltdown Scenario™ and Uncertainty Cushion™ Constraints

RDE 323 employs novel risk measures for assessing the downside risk of aportfolio. Catastrophic Meltdown Scenario™ or CMS is a weighted andsummed worst draw-down from each manager based on the worst 1 yearrolling returns. Uncertainty Cushion™ or UC provides a measure of theexpected performance of a portfolio. UC is defined as the average returnfor the portfolio minus three times its standard deviation. There is a0.5% probability that the targeted returns on the portfolio will be lessthan the Uncertainty Cushion™.

RDE 323 further permits use of these risk measures as constraints on theoptimization. Say, for a risk—averse investor who could never tolerate a10% loss even in the event of a catastrophe in the major markets, wecould devise a portfolio with an additional constraint that the CMS begreater than −10% and/or the uncertainty cushion be greater than −10%.

The constraint for the CMS is a linear constraint that can be writtenas:

${{\sum\limits_{i}{x_{i} \cdot D_{i}}} \geq {CMS}},$where D_(i) denotes the worst 1-year drawdown for asset i.

The constraint for the uncertainty cushion is non-linear constraintgiven by:μ_(P)−3σ_(P) ≧UC,where μ_(P) and σ_(P) are the mean and standard deviations as calculatedfor the portfolio respectively.

Objective Functions Employed in the Improved Resource Allocation System:FIG. 10

In the resource allocation system described in U.S. Ser. No. 10/018,696,the only objective function which could be used in optimization was theBlack-Scholes formula and the only volatility function that could beemployed in the Black-Scholes formula was the standard deviation. Theimproved resource allocation system permits the user to choose among anumber of different objective functions, to adjust the selectedobjective function for non-normal distribution of asset returns, and toselect the volatility function employed in the Black-Scholes formulafrom a number of different volatility functions. The graphical userinterface for selecting among the objective functions is shown at 1001in FIG. 10. When the user clicks on button 413, window 1003 appears.Window 1003 contains a list of the available and currently-selectableobjective functions that are available for use in basic RDE 325 androbust RDE 327. The user may select one objective function from thelist. Information about the selected objective function appears in thewindow at 1005 and the label on button 413 indicates thecurrently-selected objective function. As may be seen from the list inwindow 1003, selection of the objective function includes selection ofrobust or non-robust optimization.

The Objective Functions

The objective functions supported in the preferred embodiment are thefollowing:

Black-Scholes

The volatility and minimum return of the underlying asset and theduration of the investment horizon are used to calculate a set of optionvalues for the assets used in optimization. These option values are usedas linear objective function when optimizing inside the confidencebounds imposed by the global target portfolio return. This approach isthe one described in U.S. Ser. No. 10/018,696.

Sharpe Ratio

The expected returns, volatilities and correlations are used in aclassic non-linear maximization of the Sharpe ratio within theconfidence bounds imposed by the global target portfolio return.

Rolling Sortino Ratio

The expected returns and minimum target returns on each assets is usedin conjunction with asset volatilities and correlations to devise anon-linear objective function that measures risk-adjusted portfolioreturn in excess of the weighted minimum returns. This approach may bethought of as a Sortino ratio with a ‘moving’ Sortino target. Thisapproached is formally called the ‘Hunter Estimator’ in the userinterface, where the ‘Hunter Estimator’ represents the rolling SortinoRatio. This approach is not to be confused with the Hunter Ratioapproach described below.

Modified Black Scholes (Rolling Sortino Ratio)

The volatility in the classic Black-Scholes equation is replaced by amodified Black-Scholes volatility given by the rolling Sortino ratio orthe ‘Hunter Estimator’ (ratio of the difference between expected returnand minimum return to the asset volatility). This gives a set ofmodified Black-Scholes option values that are used as weights in alinear objective function.

Hunter Ratio

The Hunter Ratio for each asset in the optimization is computed ( as theratio of the mean of rolling Sharpe ratios to their standard deviation)and used as weights in a linear objective function that operates in thebounds of the confidence constraint imposed by the global targetportfolio return.

Modified Black Scholes (Hunter Ratio)

The volatility in the classic Black-Scholes equation is replaced by amodified Black Scholes volatility given by the Hunter Ratio of theasset/manager. This gives a set of modified Black-Scholes option valuesthat are used as weights in a linear objective function.

Adjustments to the Objective Functions

The improved asset allocation system permits a number of adjustments tothe objective function to deal with special situations that affect thedistribution of the asset returns. Among these non-normal distributionsare the effect of the degree of liquidity of the asset, the reliabilityof the returns data, and the tax sensitivity of the assets.

Adjustments for Non-normality of Returns

Non-normality of returns in the preferred embodiment may be described bykurtosis and skewness or by omega. When the non-normality described bythese measures is positive for the asset, the user manually assigns apremium to the asset's real option value; when the non-normality isnegative, the user manually assigns a discount to the asset's realoption value. Determination of skewness, kurtosis, and omega for anasset is done using the Profiler module.

Skewness and Kurtosis

Skewness is the degree of asymmetry of a distribution. In other words,it is an index of whether data points pile up on one end of thedistribution. Several types of skewness are defined mathematically. TheFisher skewness (the most common type of skewness, usually referred tosimply as “the” skewness) is defined by

${\gamma_{1} = \frac{\mu_{3}}{\mu_{2}^{3/2}}},$where μ_(i) is the ith central moment.

Kurtosis measures the heaviness of the tails of the data distribution.In other words, it is the degree of ‘peakedness’ of a distribution.Mathematically, Kurtosis is a normalized form of the fourth centralmoment of a distribution (denoted γ₂) given by

${\gamma_{2} \equiv {\frac{\mu_{4}}{\mu_{2}^{2}} - 3}},$where μ_(i) is the ith central moment. Risk-averse investors preferreturns distributions with non-negative skewness and low kurtosis.

Omega

Another measure which may be used in RDE 323 to describe non-normaldistributions is omega (Ω). Omega is a statistic defined in Con Keating& William F Shadwick, ‘A Universal Performance Measure’ (2002), TheFinance Development Centre, working paper. This is a very intuitivemeasure that allows the investor to specify the threshold between goodand bad returns and based on this threshold, identify a statistic omegaas the ratio of the expected value of returns in the “good” region overexpected value of returns in the “bad” region. Assuming, any negativereturns are unacceptable, omega is defined as

$\Omega = \frac{{Expected}\mspace{14mu}{returns}\mspace{14mu}{given}\mspace{14mu}{returns}\mspace{14mu}{are}\mspace{14mu}{positive}}{{Expected}\mspace{14mu}{returns}\mspace{14mu}{given}\mspace{14mu}{returns}\mspace{14mu}{are}\mspace{14mu}{negative}}$

Now, we can sweep the loss threshold from −∞ to ∞ and plot the statisticΩ versus the loss threshold. Comparing the Ω plot of two portfolios forrealistic loss thresholds helps us determine the superior portfolio—theone with a higher Ω for realistic loss thresholds as defined by theinvestor's risk preferences.

RDE 323 scales Ω values for an asset against an average Ω statisticusing a novel scaling mechanism depending upon the average Ω statisticand investor risk preferences and then incorporates the scaled valueinto the objective function as an option premium or discount. Omegavalues are calculated for each asset using the method described aboveand based on investor's risk preferences. Then the geometric mean ofomegas of all assets is calculated and all asset omega scaled by thismean. Any value over one gives the option premium (scaled value −1) tobe added to the asset real option value and any value less than onegives the option discount (1-scaled value) to be subtracted from thereal option value of the asset.

Adjustments for the Nature of an Asset'Liquidity

In the resource allocation system described in U.S. Ser. No. 10/018,696,the objective function did not take into account properties of theliquidity of an asset. RDE 323 has two sets of measures of liquidity: astandard measure and measures for crisis times.

The Standard Liquidity Measure

For publicly traded assets (e.g. stocks), liquidity can be quantified interms of average and lowest volume as a fraction of outstandingsecurities, average and lowest market value traded as a fraction oftotal market value, market depth for the security, derivativesavailable, open interest and volume of corresponding derivativesecurities. RDE 323 uses a novel regression model to come up with ameasure of liquidity for an asset based on relevant factors discussedabove. The model is a linear multi-factor linear regression modelwherein the coefficients of linear regression are derived using asoftware component from Entisoft (Entisoft Tools)

Crisis Liquidity Measures

The standard liquidity measure can be ineffective in times of crisiswhen there may be an overall liquidity crunch in the broad market. RDE323 defines two novel measures of liquidity that specifically addressthis concern of plummeting liquidity in times of crises:

Elasticity of Liquidity™ is the responsiveness of the measure ofliquidity of an asset to an external factor such as price or a broadmarket index. For example, an asset with elastic liquiditycharacteristics would preserve liquidity in times of crisis. On theother hand, an asset with inelastic liquidity would become illiquid andtherefore worthless during a liquidity crunch.

Velocity of Liquidity™ is the speed with which liquidity is affected asa function of time during a liquidity crisis. A measure of the velocityis the worst peak to trough fall in volume traded over the time takenfor this decline in liquidity.

RDE 323 incorporates both Elasticity of Liquidity™ and Velocity ofLiquidity™ into the objective function by means of option premiums ordiscounts that have been scaled for an average measure of liquidity andvelocity for the assets considered in the portfolio.

Liquidity of Assets such as Hedge Funds

With assets such as hedge funds, it is difficult to quantify liquidityas described above, since most of the securities data is abstracted fromthe investor and composite trading volume numbers reported at best. Insuch cases, RDE 323 determines the average liquidity of the hedge fundportfolio from the percentage of liquid and marketable assets in thehedge fund portfolio, percentage positions as a fraction of average andlowest trading volume, days to liquidate 75%/90%/100% of the portfolio,and any other liquidity information which is obtainable from the hedgefund manager. The average liquidity of the portfolio is then used todetermine an option premium or discount based and the option premium ordiscount is used as an additive adjustment to the real option value.

Adjustments for the Length of Time an Asset has been Available

RDE 323 applies reliability premiums and discounts to the objectivefunction to adjust for the length of time an asset has been available.The premium or discount is based on the “years since inception” of theasset and is a sigmoidal plot starting out flat till 2-3 years, thenincreasing steadily through 7-8 years and then flattening out slowly as“years since inception” increase even further. Another way of dealingwith assets for which long-term information is not available is to makescenarios for the portfolio that contains them and apply robust RDE 327to the portfolio as described above.

Adjustments for the Tax Sensitivity of an Asset

The ultimate returns from an asset which are received by the investorare of course determined by the manner in which the returns are taxed.Returns from tax-exempt assets, from tax-deferred assets, and returns inthe forms of dividends, long-term gains, and short-term gains are taxeddifferently in many taxation systems. In RDE 323, the expected returnsand covariance of the assets are calculated post-tax assuming taxefficiency for the asset and tax criteria of the account considered.During optimization, the post-tax inputs are used in the objectivefunction and in the constraints.

Tax sensitivity of an asset can be gauged by the following threeparameters that are reported by funds/managers:

Turnover,

$T = \frac{{Realized}\mspace{14mu}{Returns}}{{Total}\mspace{14mu}{{Reported}\left( {{Realized} + {Unrealized}} \right)}}$

Long-Term/Short-Term Cap-Gains,

$R_{LS} = \frac{{Long}\text{-}{Term}\mspace{14mu}{Capital}\mspace{14mu}{Gains}}{{Short}\text{-}{Term}\mspace{14mu}{Capital}\mspace{14mu}{Gains}}$

Dividends, D=Dividend Yield

Let the tax rates on long-term cap-gains, short-term cap gains anddividends be i_(l), i_(S) and i_(D) respectively. These rates can becustomized for each client and account as described below. Thetax-modified returns for the manager are then given byr _(tax-modified)=[(1−T)+(T−D)[R _(LS)(1−i _(L))+(1−R _(LS))(1−i_(S))]+D(1−i _(D))]r _(reported)

For example, if the turnover for some manager is 30% and the ratio oflong-term to short-term cap gains is 40% with a dividend of 2%, thenwith taxes rates 18% for long-term cap gains and dividends and 38% forshort-term cap gains, the tax-modified returns would be 91% of thereported returns.

The relative tax-efficiency of the manager can be assessed by thetax-efficiency factor that is given by

${{Tax}\mspace{14mu}{Efficiency}} = \frac{\begin{matrix}{1 - \left\lbrack {\left( {1 - T} \right) + {\left( {T - D} \right)\left\lbrack {{R_{LS}\left( {1 - i_{L}} \right)} +} \right.}} \right.} \\\left. {\left. {\left( {1 - R_{LS}} \right)\left( {1 - i_{S}} \right)} \right\rbrack + {D\left( {1 - i_{D}} \right)}} \right\rbrack\end{matrix}}{T}$

For the hypothetical manager considered above, Tax Efficiency would be0.3. As can be seen from the expression above, the tax efficiency of anasset increases with increases in the fraction of long-term capitalgains in the realized returns. Less turnover also increases the asset'stax efficiency. This can be explained by the fact that as turnoverdecreases, the percentage of the gains that are realized as long-termgains increases.

A simpler measure of tax sensitivity has been devised for investmentmanagement applications. In this measure, reported returns are assumedto be made up of realized capital gains (long-term and short-term),income (dividends), and unrealized capital gains. Post-tax returns arefound by deducting the respective taxes on long and short-term capitalgains and dividends from the reported returns. The asset module is usedto associate the information needed to determine tax efficiency with theasset.

Customizable Client Tax Rates

The tax rates for each client/account can be customized according towhether the account is tax-exempt, tax-deferred or otherwise. State taxand alternative minimum tax rates can be imposed via specifying thelong-term, short-term and dividend tax rates. These tax rates are themused to calculate the post-tax returns and covariance for the assets inthe portfolio.

Options for Quantifying an Asset's Risk

RDE 323 offers the user three modes of quantifying the risk of an asset.RDE 323 then uses the risk as quantified according to the selected modeto calculate the real option values. The modes are:

-   1. Flat Risk: The flat risk assumes a uniform risk (say −10%) on    each asset in the portfolio.-   2. Mean—2* Standard Deviation: Another commonly used measure of the    risk of investing in an asset is the mean minus twice the standard    deviation of the returns distribution on an asset. Statistically,    there is a 5% probability of the returns falling below this measure    (assuming a normal distribution of returns for the asset)-   3. Worst 1-year rolling return: This is a conservative estimate of    the risk associated with investing in an asset. It measures risk as    the worst 1-year rolling return on the asset since its inception.

Implementation Details of a Preferred Embodiment: FIGS. 11-12

The improved asset allocation system is implemented with a GUI createdusing Microsoft Visual Basic. Microsoft COM and .NET compliantcomponents, Excel Automation for report generation, a Matlaboptimization engine for numerical computations and optimization support,and a robust back-end SQL Server database for data storage. MicrosoftVisual Basic, Excel Automation, and SQL Server are all manufactured byMicrosoft Corporation of Richmond, Wash. The Matlab optimization engineand the programs that perform the computations are part of the Matlabprogram suite available from The Math Works, Inc., Natick, Mass.Microsoft, Visual Basic, Excel, and SQL Server are trademarks ofMicrosoft Corporation; Matlab and The Math Works are trademarks of TheMath Works, Inc. FIG. 11 is a functional block diagram of improved assetallocation system 1109. User 1103 interacts with system 1101 via VisualBasic programs 1105. Data describing assets, portfolios, and parametersfor optimizations, as well as the results of the optimizations iswritten to and read from the database in SQL Server back end 1107, whilethe mathematical computations are performed by optimization engine 1109,which is thus an implementation of RDE 323.

Details of the SQL Server Database: FIG. 12

FIG. 12 shows the tables in relational database 1201 in SQL Server 1107.For purposes of the present discussion, the tables fall into fourgroups:

-   -   account tables 1203, which contains a single table, account        table 1205, which contains information about the accounts for        which asset allocation optimizations are made.    -   Report tables 1206, which contain information needed to prepare        reports.    -   Asset tables 1211, which contain asset-related information; and    -   Optimization run tables 1221, which contain information related        to optimizations of portfolios of assets by RDE 323.

The tables that are of primary importance in the present context areasset tables 1211 and optimization run tables 1221.

Each optimization run of RDE 323 is made for an account on a set ofassets. The run uses a particular objective function and applies one ormore constraints to the: optimization. Tables 1203, 1211, and 1221relate the account, the set of assets, and the constraints to the run.Beginning with accounts table 1205, there is one entry in accounts table1205 for each account; of the information included in the entry for anaccount, the identifier for the entry and the tax status information forthe account is of the most interest in the present context. The entryspecifies whether the account is tax deferred, the account's long termcapital gains tax rate, and its short term capital gains tax rate.

Asset Tables 1211

Tables 1211 describe the assets. The main table here is assets table1217, which has an entry for each kind of asset or benchmark used in RDE323. Information in the entry which is of interest in the presentcontext includes the identifier for the asset, information that affectsthe reliability of information about the asset, and informationconcerning the percentage of the yields of the asset come from long-termand short-term gains and the dividend income. RDE 323 keeps differentinformation for an entry in asset table 1217 depending on whether itrepresents an asset or a benchmark. When the entry is an asset, theextra information is contained in investment table 1215. There is anentry in investment table 1215 for each combination of asset andaccount. When the entry is a benchmark, the extra information iscontained in BenchMarkAsset table 1211, which relates the asset to thebenchmark. AssetReturns table 1213, finally, relates the asset to thecurrent return information for the asset. This information is loadedfrom current market reports into asset returns table 1213 prior to eachoptimization by RDE 323.

Optimization Run Tables 1221

The chief table here is RDERun table 1223. There is an entry in RDERuntable 1223 for each optimization run that has been made by RDE 323 andnot deleted from the system. The information in an RDERun table entryfalls into two classes: identification information for the run andparameters for the run. The identification information includes anidentifier, name, and date for the run, as well as the identifier forthe record in account table 1205 for the client for which the run wasmade.

Parameters include the following:

-   -   Parameters for defining the optimization, including the start        date and end date for the historical data about the assets, the        anticipated rate for risk-free investments, and the investment        horizon.    -   The mode by which the risk is to be quantified;    -   The minimum return desired for the portfolio    -   The range of returns for which a confidence value is desired;    -   The optimization method (i.e., the objective function to be        employed in the optimization);    -   Tax rate information for the run;    -   the number of multiple asset constraints for the run;    -   Constraints based on the return, risk, Sharpe Ratio, tax        efficiency, and reliability for the optimized portfolio.

One or more RDEMMConstraintAssets entries in RDEMMConstraintAssets table1225 may be associated with each RDERun entry. EachRDEMMConstraintAssets entry relates the RDERun entry to one of a set ofconstraints that apply to multiple assets. RDERunAssets table 1227,finally, contains an entry for each asset-run combination. For aparticular run and a particular asset that belongs to the portfoliooptimized by the run, the entry indicates the initial weight of theasset in the portfolio being optimized in the run, any constraints forthe minimum and maximum weights permitted for the asset in the portfoliobeing optimized, and the weight of the asset in the portfolio asoptimized by the run.

When database schema 1201 is studied in conjunction with thedescriptions of the graphical user interfaces for inputting informationinto RDE 323, the descriptions of the optimization operations, and thedescriptions of the effects of the constraints on the optimizationoperations, it will be immediately apparent to those skilled in therelevant technologies how system 1101 operates and how a user of system1101 may easily define different portfolios of assets, may select assetsfor a portfolio according to the MMF reliability of the set of assets,and may optimize the portfolio to obtain a weighting of the assets inthe portfolio that is made according to the real option values of theassets as constrained by a total return reliability constraint. Theoptimization may be done using either standard optimization techniquesor robust optimization techniques. A user of system 1101 may with equalease make various adjustments to the objective function used to computethe real option values of the portfolio's assets and may also subjectthe optimization to many constraints in addition to the total returnreliability constraint.

CONCLUSION

The foregoing Detailed Description has disclosed to those skilled in therelevant technologies how to make and use the improved resourceallocation system in which the inventions disclosed herein are embodiedand has also disclosed the best mode presently known to the inventors ofmaking the improved resource allocation system. It will be immediatelyapparent to those skilled in the relevant technologies that theprinciples of the inventions disclosed herein may be used in ways otherthan disclosed herein and that resource allocation systems incorporatingthe principles of the invention may be implemented in many differentways. For example, the principles disclosed herein may be used toallocate resources other than financial assets. Further, the techniquesdisclosed herein may be used with objective functions, constraints onthe objective functions, and adjustments to the objective functionswhich are different from those disclosed herein, as well as withscenarios for robust optimization which are different from the onesdisclosed herein. Finally, many different actual implementations ofresource allocation systems that incorporate the principles of theinventions disclosed herein may be made. All that is actually requiredis a store for the data and a processor that has access to the store andcan execute programs that generate the user interface and do themathematical computations. For example, an implementation of theresource allocation system could easily be made in which the computationand generation of the user interface was done by a server in the WorldWide Web that had access to financial data stored in the server orelsewhere in the Web and in which the user employed a Web browser in hisor her PC to interact with the server.

For all of the foregoing reasons, the Detailed Description is to beregarded as being in all respects exemplary and not restrictive, and thebreadth of the invention disclosed herein is to be determined not fromthe Detailed Description, but rather from the claims as interpreted withthe full breadth permitted by the patent laws.

1. A method of analyzing a reliability of a set of more than one asset,the assets in the set being selected from a plurality of assets,historic returns data for the assets of the plurality being stored instorage accessible to a processor, and the method comprising the stepswhich the processor has been programmed to perform of: receiving inputsindicating assets selected for the set and for each selected asset, adesired minimum return; using the historic returns data to determine aprobability that at least one of the assets in the set will not providethe desired minimum return indicated for the asset; and outputting theprobability as an indication of the reliability of the set of assets. 2.The method set forth in claim 1 wherein the step of using the historicreturns to determine a probability comprises the steps of: using themultivariate normal distribution for the returns of the assets todetermine the probability that each of the selected assets will providethe desired minimum return; and determining the probability that atleast one of the selected assets will not provide the desired minimumreturn from the probability that each of the selected assets willprovide the desired minimum return.
 3. The method set forth in claim 2wherein: in the step of using the multivariate normal distribution, theprobability that each of the selected assets will provide the desiredreturn is determined using real option values of the assets.
 4. Themethod set forth in claim 1 wherein: the received inputs include aperiod of time; and the probability is the probability over the periodof time.
 5. A method of optimizing a value of a set of assets over a setof a plurality of scenarios, each scenario in the set of scenariosaffecting values of assets in the set of assets, historic returns datafor the assets being stored in storage accessible to a processor, andthe method comprising the steps which the processor has been programmedto perform of: receiving inputs indicating the set of scenarios, eachscenario specifying values which are used in optimizing the set ofassets and which vary stochastically between two extremes and eachscenario specifying a probability of occurrence for the scenario; andoptimizing weights of the assets in the set to maximize a worst-casevalue of the set of assets over the set of scenarios.
 6. The method ofoptimizing set forth in claim 5 wherein: the worst-case value of the setof assets is the worst-case real option value thereof; and the valueswhich are used in optimizing are the mean return and the covariance. 7.The method of optimizing set forth in claim 5 wherein: a scenario in theset of scenarios may correspond to the historical returns data for theassets in the set of assets.
 8. The method of optimizing set forth inclaim 5 wherein: a scenario in the set of scenarios may include certainassets in the set of assets which are highly correlated.
 9. The methodof optimizing set forth in claim 5 wherein: a scenario in the set ofscenarios may correspond to outliers in the historical returns data. 10.The method of optimizing set forth in claim 5 further comprising thestep of: receiving inputs indicating additional constraints to which theset of assets being optimized is subject; and in the step of optimizingweights of the assets, optimizing the weights subject to the additionalconstraints.
 11. A method of selecting a set of assets from a pluralitythereof of the sets of assets and optimizing maximizing the weights ofthe assets in a value of the selected set of assets, historic returnsdata for assets being stored in storage accessible to a processor andthe method comprising the steps performed in the which the processor hasbeen programmed to perform of: 1) selecting a the set of assets on thebasis of a reliability of the set of assets, the reliability being aprobability based on the historic returns data that at least one assetof the set of assets in a selected set will not provide a desiredminimum return indicated specified for the asset; and 2) optimizing theweights of the assets in the selected set of assets to maximize thevalue of the selected set of assets.
 12. The method set forth in claim11 wherein: the probability that at least one of the assets will notprovide the desired minimum return is determined using the real optionvalues for the assets.
 13. The method set forth in claim 11 wherein:optimizing the weights of the assets in the selected set of assets isdone using the real option values for the assets.
 14. The method setforth in claim 13 wherein: the method further includes the step of:receiving an input indicating one of a plurality of objective functionsfor computing the real option values for the assets; and in the step ofoptimizing the weights of the assets in the selected set of assets, theoptimization is done using the indicated objective function of theplurality.
 15. The method set forth in claim 14 wherein: in the step ofoptimizing the weights of the assets, the objective function is adjustedby assigning a premium or a discount to the real option value of one ormore of the assets.
 16. The method set forth in claim 15 wherein: theobjective function is adjusted to take non-normal returns for the assetinto account.
 17. The method set forth in claim 15 wherein: theobjective function is adjusted to take liquidity characteristics of theasset into account.
 18. The method set forth in claim 15 wherein: theobjective function is adjusted to take tax sensitivity of an asset intoaccount.
 19. The method set forth in claim 15 wherein: the objectivefunction is adjusted to take the length of time an asset has beenavailable into account.
 20. The method set forth in claim 13 wherein:the method further includes the step of: receiving an input indicatingone of a plurality of modes of quantifying the risk of an asset; and inthe step of optimizing the weights of the assets in the selected set ofassets, the optimization is done using the indicated mode of theplurality.
 21. The method set forth in claim 11 wherein: optimizing theweights of the assets in the selected set of assets is done using robustoptimization.
 22. The method set forth in claim 21 wherein: the robustoptimization optimizes over a set of user-specified scenarios, eachscenario having values which are used in optimizing the selected set ofassets and which vary stochastically between two extremes and aprobability of occurrence for the scenario.
 23. The method set forth inclaim 11 wherein: optimizing the weights of the assets is done subjectto a constraint that the probability that the selected set of assetsyields a desired minimum return is greater than a user-specified valuea.
 24. The method set forth in claim 23 wherein: the optimization isdone subject to a plurality of constraints (1 . . . n) , a constraintc_(i) specifying that the probability that the selected set of assetsyields a desired minimum return that is greater than a user-specifiedvalue a _(i).
 25. The method set forth in claim 23 wherein: optimizingthe weights of the assets in the selected set of assets is done usingrobust optimization.
 26. The method set forth in claim 25 wherein: therobust optimization optimizes over a set of user-specified scenarios,each scenario including a mean return and a covariance matrix, each ofwhich varies stochastically between two extremes, and a probability ofoccurrence for the scenario.
 27. The method set forth in claim 11wherein: the an asset in the selected set of assets may have a negativeweight.
 28. The method set forth in claim 11 wherein; the sum of theweights of the assets in the selected set of assets may exceed
 1. 29.The method set forth in claim 11 wherein: optimizing the weight of theassets in the selected set of assets is done subject to one or moreadditional constraints.
 30. The method set forth in claim 29 wherein:the additional constraint restricts the sum of the weights of the assetsbelonging to a selected subset of the assets in the selected set ofassets.
 31. The method set forth in claim 29 wherein: the additionalconstraint constrains the weight of an asset such that the amount of theasset in the selected set of assets is above a minimum investmentthreshold.
 32. The method set forth in claim 29 wherein: the additionalconstraint limits constrains the set's downside risk of the selected setof assets to be less than a predetermined value b.
 33. The method setforth in claim 32 wherein; the additional constraint is computed fromthe worst draw-down for each asset.
 34. The method set forth in claim 32wherein: the additional constraint is computed from the set's averagereturn and standard deviation for the selected set of assets.
 35. Themethod set forth in claim 11 wherein: the probability is the probabilityover a period of time.